Séminaires & Congrès
14, 2006, p. 65–82
PAINLEVÉ PROPERTY OF THE HÉNON-HEILES
HAMILTONIANS
by
Robert Conte, Micheline Musette & Caroline Verhoeven
Abstract. — Time independent Hamiltonians of the physical type
H = (P12 + P22 )/2 + V (Q1 , Q2 )
pass the Painlevé test for only seven potentials V , known as the Hénon-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painlevé
property was not yet achieved for generic values of the free constants. We integrate
each missing case by building a birational transformation to some fourth order first
degree ordinary differential equation in the classification (Cosgrove, 2000) of such
polynomial equations which possess the Painlevé property. The properties common
to each Hamiltonian are:
(i) the general solution is meromorphic and expressed with hyperelliptic functions
of genus two,
(ii) the Hamiltonian is complete (the addition of any time independent term would
ruin the Painlevé property).
Résumé (Propriété de Painlevé des hamiltoniens de Hénon-Heiles). — Les hamiltoniens,
indépendants du temps, de la forme
H = (P12 + P22 )/2 + V (Q1 , Q2 )
satisfont au test de Painlevé pour seulement sept potentiels V ; ceux-ci sont connus
sous le nom de hamiltoniens de Hénon-Heiles et ils dépendent d’un nombre fini de
constantes libres. La propriété de Painlevé restait à établir pour des valeurs génériques
des constantes libres. Nous traitons chacun des cas en suspens en construisant une
transformation birationnelle vers une équation différentielle ordinaire d’ordre quatre
qui figure dans la liste exhaustive (Cosgrove, 2000) de telles équations polynomiales
possédant la propriété de Painlevé. Les propriétés communes à ces hamiltoniens sont :
(i) la solution générale est méromorphe et peut être exprimée en termes de fonctions hyperelliptiques de genre deux,
(ii) le hamiltonien est complet au sens où l’addition de tout terme indépendant
du temps ferait perdre la propriété de Painlevé.
2000 Mathematics Subject Classification. — Primary 34M60; Secondary 34E20, 34M55, 34M35.
Key words and phrases. — Hénon-Heiles Hamiltonian, Painlevé property, hyperelliptic functions, separation of variables, Darboux coordinates.
The authors acknowledge the financial support of the Tournesol grant no. T2003.09 between Belgium
and France. C. Verhoeven is a postdoctoral fellow at the FWO-Vlaanderen.
c Séminaires et Congrès 14, SMF 2006
66
R. CONTE, M. MUSETTE & C. VERHOEVEN
1. Introduction
Let us consider the most general two-degree of freedom, classical, time-independent
Hamiltonian of the physical type (i.e, the sum of a kinetic energy and a potential
energy),
1 2
(1)
(p + p22 ) + V (q1 , q2 ),
H =
2 1
and let us require that the general solution q1n1 , q2n2 , with n1 , n2 integers to be determined, be single valued functions of the complex time t, i.e., what is called the
Painlevé property of these equations.
A necessary condition is that the Hamilton equations of motion, when written in
these variables q1n1 , q2n2 , pass the Painlevé test ([12]). This selects seven potentials V
(three “cubic” and four “quartic”) depending on a finite number of arbitrary constants,
which are known as the Hénon-Heiles Hamiltonians ([24]). In order to prove the
sufficiency of these conditions, one must then perform the explicit integration and
check the singlevaluedness of the general solution. We present here a review on this
subject.
The paper is organized as follows:
In section 2, we enumerate the seven cases isolated by the Painlevé test, together
with the second constant of the motion K in involution with the Hamiltonian. In
section 3, we recall the separating variables in the four cases where they are known.
In section 4, we display confluences from quartic cases to all the cubic cases, thus
restricting the problem to the consideration of the quartic cases only. In section
5, due to the lack of knowledge of the separating variables in the three remaining
cases, we state the equivalence of the equations of motion and the conservation of
energy with some fourth order first degree ordinary differential equations (ODEs).
In section 6, since these fourth order equations do not belong to any set of already
classified equations, we build a birational transformation between each quartic case
and some fourth order ODE belonging to a classification of Cosgrove ([17]), thus
proving the Painlevé property for the quartic cases.
To summarize, the results are twofold:
1. each case is integrated by solving a Jacobi inversion problem involving a hyperelliptic curve of genus two, which proves the meromorphy of the general
solution,
2. each case is complete in the sense of Painlevé, i.e, it is impossible to add any timeindependent term to the Hamiltonian without ruining the Painlevé property.
2. The seven Hénon-Heiles Hamiltonians
By application of the Painlevé test, one isolates two classes of potentials V (q1 , q2 ),
called “cubic” and “quartic” for simplification.
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
67
1. In the cubic case HH3 ([10, 13, 21]),
(2)
H=
1 2
1
1
(p + p22 + ω1 q12 + ω2 q22 ) + αq1 q22 − βq13 + γq2−2 ,
2 1
3
2
α 6= 0,
in which the constants α, β, ω1 , ω2 and γ can only take the three sets of values,
(3)
(SK) :
(4)
(KdV5) :
(5)
β/α = −1, ω1 = ω2 ,
β/α = −6,
(KK) : β/α = −16, ω1 = 16ω2 .
2. In the quartic case HH4 ([23, 32]),
(6) H =
(7)
1 2
(P + P22 + Ω1 Q21 + Ω2 Q22 ) + CQ41
2 1
1 α
β
2 2
4
+ BQ1 Q2 + AQ2 +
+ 2 + γQ1 ,
2 Q21
Q2
B 6= 0,
in which the constants A, B, C, α, β, γ, Ω1 and Ω2 can only take the four values
(the notation A : B : C = p : q : r stands for A/p = B/q = C/r = arbitrary),

A : B : C = 1 : 2 : 1,
γ = 0,



A : B : C = 1 : 6 : 1,
γ = 0, Ω1 = Ω2 ,

A
:
B
:
C
=
1
:
6
:
8,
α = 0, Ω1 = 4Ω2 ,


A : B : C = 1 : 12 : 16, γ = 0, Ω1 = 4Ω2 .
For each of the seven cases so isolated there exists a second constant of the motion
K ([7, 18, 25]) ([6, 7, 26]) in involution with the Hamiltonian,
(SK)
K = 3p1 p2 + αq2 (3q12 + q22 ) + 3ω2 q1 q2
2
+ 3γ(3p21 q2−2 + 4αq1 + 2ω2 ),
(KdV5) K = 4αp2 (q2 p1 − q1 p2 ) + (4ω2 − ω1 )(p22 + ω2 q22 + γq2−2 )
+ α2 q22 (4q12 + q22 ) + 4αq1 (ω2 q22 − γq2−2 ),
(KK)
K = (3p22 + 3ω2 q22 + 3γq2−2 )2 + 12αp2 q22 (3q1 p2 − q2 p1 )
− 2α2 q24 (6q12 + q22 ) + 12αq1 (−ω2 q24 + γ) − 12ω2 γ,



α
β

 K = (Q2 P1 − Q1 P2 )2 + Q22 2 + Q21 2


Q1
Q2


Ω
−
Ω
α
β
(1 : 2 : 1)
1
2
2
2
4
4
2
2
−
P
−
P
+
Q
−
Q
+
Ω
Q
−
Ω
Q
+
−
,

1
2
1
2
1
2
1
2


2
Q21
Q22

1



 A = 2,
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
68
R. CONTE, M. MUSETTE & C. VERHOEVEN



Q2 + Q2
2



1
2

−
K
=
P
P
+
Q
Q
+
Ω
1 2
1 2
1


8



κ2
κ2 κ2
κ2
1 2 2
(1 : 6 : 1)
− P22 12 − P12 22 +
κ1 Q2 + κ22 Q21 + 12 22 ,


Q1
Q2
4
Q1 Q2




1


α = −κ21 , β = −κ22 , A = − ,



32


2


Q22
β

2
2
2


K = P2 −
(2Q2 + 4Q1 + Ω2 ) + 2


16
Q2





1 2

2
− Q2 (Q2 P1 − 2Q1 P2 ) + γ − 2γQ22 − 4Q2 P1 P2
4
(1 : 6 : 8)



1
β

4
3 2
2
2

+ Q1 Q2 + Q1 Q2 + 4Q1 P2 − 4Ω2 Q1 Q2 + 4Q1 2 ,



2
Q2

1



A
=
−
,


16





K
=
8(Q2 P1 − Q1 P2 )P2 − Q1 Q42 − 2Q31 Q22




2
Q22 P22
32α
β
(1 : 12 : 16)
4
2
,
Q
+
10
+
+
2Ω
Q
Q
−
8Q
1 1 2
1 2

2

Q2
5
Q21


1



 A = − 32 .,
Remark. — Performing the reduction q1 = 0, p1 = 0 in the three HH3 Hamiltonians (2) yields H = p2 /2 + (1/2)ωq 2 + (1/2)γq −2 , for which q 2 obeys a linearizable
Briot-Bouquet ODE. Similarly, the reduction Q1 = 1, P1 = 0 in the four HH4 Hamiltonians (6) yields H = P 2 /2 + (1/2)ωQ2 + AQ4 + (1/2)βQ−2, for which Q2 obeys the
Weierstrass elliptic equation.
These seven Hénon-Heiles Hamiltonians can be studied from various points of view
such as: separation of variables ([37]), Painlevé property, algebraic complete integrability ([3]). For the interrelations between these various approaches, the reader can
refer to the plain introduction in Ref. [1]. In the present work, we only deal with
proving the Painlevé property (PP).
In order to prove or disprove the PP, it is sufficient to obtain an (explicit) canonical
transformation to new canonical variables (the so-called separating variables) which
separate the Hamilton-Jacobi equation for the action S(q1 , q2 ) ([5, chap. 10]), which
for two degrees of freedom is
∂S
∂S
(8)
H(q1 , q2 , p1 , p2 ) − E = 0, p1 =
, p2 =
.
∂q1
∂q2
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
69
Indeed, if such separating variables are obtained, depending on the genus g of the
hyperellitic curve r2 = P (s) involved in the associated Jacobi inversion problem,
(9)
ds
ds
p 1 + p 2 = 0,
P (s1 )
P (s2 )
s ds
s ds
p1 1 + p2 2 = dt,
P (s2 )
P (s2 )
the elementary symmetric functions s1 +s2 and s1 s2 are either meromorphic functions
of time (g ≤ 2), or multivalued (g > 3).
3. The four cases with known separating variables
Two of the seven cases (KdV5, 1:2:1) have a second invariant K equal to a second
degree polynomial in the momenta, therefore there exists a classical method ([38,
39]) to obtain the canonical transformation (q1 , q2 , p1 , p2 ) → (s1 , s2 , r1 , r2 ) with the
separating variables (s1 , s2 ) obeying the canonical system (9). For the KdV5 case,
one obtains ([4, 18, 45])

q1 = −(s1 + s2 + ω1 − 4ω2 )/(4α), q22 = −s1 s2 /(4α2 ),




2

s1 r1 − s2 r2

2
2 s1 s2 (r1 − r2 )

,
p
=
−16α
,
p
=
−4α

1
2

s1 − s2
(s1 − s2 )2





f (s1 , r1 ) − f (s2 , r2 )


,
 H=
s1 − s2
(10)

s2 (s + ω1 − 4ω2 )2 (s − 4ω2 ) − 64α4 γ


+ 8α2 r2 s,
f (s, r) = −

2s

32α




K


f (sj , rj ) − Esj +
= 0, j = 1, 2,



2



P (s) = s2 (s + ω1 − 4ω2 )2 (s − 4ω2 ) + 32α2 Es2 − 16α2 Ks − 64α4 γ.
For 1:2:1, one obtains



qj2 =








pj =








H =





f (s, r) =
(11)












 f (sj , rj ) =







P (s) =



(s1 + ωj )(s2 + ωj )
, j = 1, 2,
ω1 − ω2
ω3−j (r2 − r1 ) − s1 r1 + s2 r2
2qj
, j = 1, 2,
s1 − s2
f (s1 , r1 ) − f (s2 , r2 )
,
s1 − s2
s3
ω1 + ω2 2
2(s + ω1 )(s + ω2 )r2 −
−
s
2
2 α
ω1 ω2
ω2 − ω1
β −
s+
−
,
2
2
s + ω1
s + ω2
α+β
ω1 + ω2
K
−
− sj + E
− , j = 1, 2,
2
2
2
s(s + ω1 )2 (s + ω2 )2 − α(s + ω2 )2 − β(s + ω1 )2
−(s + ω1 )(s + ω2 ) [E(2s + ω1 + ω2 ) − K] .
(−1)j
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
R. CONTE, M. MUSETTE & C. VERHOEVEN
70
The two cubic cases SK and KK,
1 2
Ω1 2
1
1
λ2
(P1 + P22 ) +
(Q1 + Q22 ) + Q1 Q22 + Q31 + Q−2
,
2
2
2
6
8 2
ω2
1
4
λ2
1
(13)
(16q12 + q22 ) + q1 q22 + q13 + q2−2 ,
HKK = (p21 + p22 ) +
2
2
4
3
2
are equivalent under a birational canonical transformation ([8, 36]). Therefore, the
separating variables (s1 , s2 ) are common to these two cases.
In the nongeneric case λ = 0, the separating variables have been built ([33]) by a
method ([2, 40]) based on the local representation of the general solution q1 (t), q2 (t)
by a Laurent series of t−t0 near a movable singularity t0 . The algebraic curves defined
by the values of the two invariants H, K in terms of the arbitrary coefficients of the
Laurent series are then geometrically interpreted, with, in principle, the separating
variables as the final output. However, some technical difficulty prevents this method
to handle the generic case λ 6= 0.
The generic case can nevertheless be separated ([42]) and the result is
(12)
HSK =

!2


P̃1 − P̃2
f (Q̃1, P̃1 ) − f (Q̃2 , P̃2 )
Q̃1 + Q̃2


q1 = −6
,
q22 = 24
,
−


2

Q̃
−
Q̃
Q̃1 − Q̃2
1
2







P̃1 − P̃2
Q̃1 P̃2 − Q̃2 P̃1
P̃1 − P̃2


−2
,
p2 = Q̃2
,
p1 = −4Q̃1



Q̃1 − Q̃2
Q̃1 − Q̃2
Q̃1 − Q̃2






λ2
Q̃1 − Q̃2



H = f (Q̃1 , P̃1 ) + f (Q̃2 , P̃2 ) +
,


24
f
(
Q̃
,
P̃

1
1 ) − f (Q̃2 , P̃2 )


1
(14)
f (q, p) = p2 + q 3 − 4ω22 q,



12



2

 
E
λ2


f
(
Q̃
,
P̃
)
−
+ Q̃j + K = 0, j = 1, 2,
j
j


2
24






3K
3K
r1
r2


Q̃1 = s21 − 2 , Q̃2 = s22 − 2 , P̃1 =
, P̃2 =
,



λ
λ
2s1
2s2




3


K
λ
1 2
K


 P (s) = −
s −3 2
+ Ω21 s2 − 3 2 + √ s + 2E.
3
λ
λ
3
It is remarkable that the canonical transformation
(15)
(q1 , q2 , p1 , p2 ) −→
Q̃1 − Q̃2
Q̃1 + Q̃2
+ Ω1 ,
, P̃1 + P̃2 , P̃1 − P̃2
2
2
!
coincides with the canonical transformation between the SK variables and the KK
variables in the particular case λ = 0.
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
71
In the three remaining cases, the quartic 1:6:1, 1:6:8, 1:12:16, the separating variables are only known in nongeneric cases ([41, 43]), and the associated particular
solutions are single valued. In order to decide about the Painlevé property, which
only involves the general solution, one must therefore integrate by different means.
4. Confluences from the quartic cases to the cubic ones
A possible way to integrate would be to take advantage of some confluence from
an integrated case to a not yet integrated case. For instance, the property of single
valuedness of the general solution of the second Painlevé equation P2 implies, from
the classical confluence from P2 to P1, the same property for P1.
The confluence from the quartic 1:6:8 case to the cubic KK case found in Ref. [35]
is not an isolated feature ([41]), and in fact all the cubic cases can be obtained by a
confluence of at least one quartic case. Just like between the six Painlevé equations,
one of the parameters in the Hamiltonian is lost in the process. Consider, for instance,
the quartic 1:12:16 and the cubic KK cases,

1
ω

h1:12:16 (t) = (p21 + p22 ) + (4q12 + q22 )


2
8

1 α
β
n

+ 2 ,
− (16q14 + 12q12 q22 + q24 ) +
2
(16)
32
2 q1
q

2B


1
Ω
16

 HKK (T ) = (P12 + P22 ) + (16Q21 + Q22 ) + N Q1 Q22 + Q31 +
,
2
2
3
2Q22
The confluence in this case is

4

 t = εT, q1 = ε−1 + Q1 , q2 = Q2 , n = − ε−1 N,


3
4
1:12:16 → KK
−7
−2
N
+
4Ωε
,
β
=
ε
B,
α
=
ε
−


3

 ω = ε−3 (−4N
+ 4Ωε), h = ε−5 (−2N + 4Ωε + Hε3 ), ε → 0,
and the two quartic parameters (α, ω) coalesce to the single cubic parameter Ω.
We have checked that all the generic cubic cases can be obtained by confluence
from at least one quartic case, as indicated in the following list:


 HH4 1:2:1 → HH3 KdV5,



 HH4 1:6:8 → HH3 KK,
(17)
HH4 1:6:8 → HH3 KdV5,


 HH4 1:12:16 → HH3 KK,



HH4 1:12:16 → HH3 SK.
Since these confluences are not invertible and always go from quartic to cubic, they
are unfortunately of no help to integrate the missing cases, which are all quartic. In
section 6, we present another class of transformations, these one invertible, between
some of the seven cases, which indeed helps to integrate the missing cases.
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
72
R. CONTE, M. MUSETTE & C. VERHOEVEN
5. Equivalent fourth order ODEs
The Painlevé school has “classified” (i.e, enumerated the integrable equations and
integrated them) several types of ODEs (e.g., second order first degree, third order
first degree of the polynomial type, etc), but no four-dimensional first order differential
system such as the Hamilton equations
dqj
dpj
∂V
= pj ,
=−
, j = 1, 2
dt
dt
∂qj
(18)
has ever been classified. However, some types of fourth order ODEs have been classified, in particular the polynomial class ([9, 11, 16, 17])
u0000 = P (u000 , u00 , u0 , u, x),
(19)
in which P is polynomial in u000 , u00 , u0 , u and analytic in x. Therefore, if one succeeds,
by elimination of either q1 or q2 (or another combination) between the two Hamilton
equations and the equation H = E expressing the conservation of the energy, to build
a fourth order ODE in the class (19), and if this ODE is equivalent to the original
system, then the question is settled.
In the cubic case, the two Hamilton equations
(20)
q100 + ω1 q1 − βq12 + αq22 = 0,
(21)
q200 + ω2 q2 + 2αq1 q2 − γq2−3 = 0,
together with H − E = 0, see (2), are indeed equivalent ([21]) to the single fourth
order first degree ODE for q1 (t),
20
αβq13
3
+(ω1 + 4ω2 )q100 + (6αω1 − 4βω2 )q12 + 4ω1 ω2 q1 + 4αE = 0.
q10000 + (8α − 2β)q1 q100 − 2(α + β)q102 −
(22)
The equivalence results from the conservation of the number of parameters between
the system (20)–(21) and the single equation (22), since the coefficient γ of the nonpolynomial term q2−2 has been replaced by the constant value E of the Hamiltonian
H. The results of the classification ([17]) enumerate as expected only three Painlevéintegrable such equations and they provide their general solution (for the first time
in the SK and KK cases).
In the quartic case, the similar fourth order equation is built by eliminating Q2
2
and Q000
between the two Hamilton equations,
1
(23)
Q001 + Ω1 Q1 + 4CQ31 + 2BQ1 Q22 − αQ−3
1 + γ = 0,
(24)
Q002 + Ω2 Q2 + 4AQ32 + 2BQ2 Q21 − βQ−3
2 = 0,
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
73
and the Hamiltonian (6), which results in
2
2
Q01 Q000
A Q001
Q01 Q001
1
−Q0000
+
2
+
1
+
6
−
2
1
Q1
B Q1
Q21
AC
AC
2
+8 6
− B − C Q21 Q001 + 4(B − 2C)Q1 Q01 + 24C 4
− B Q51
B
B
A γ
A α
A
(25)
Q001
−4 1+3
+ 12 ω1 − 4ω2 + 1 + 12
B
B Q1
B Q41
2
A α2
α
Q01
α 02
A γα
A
+6
γ
−
−
2γ
+
20
Q
−
12
+
4
3
ω
−
ω
1
2
1
B Q71
Q51
B Q41
B
Q31
Q21
1
AC
A
A 2
γ + 2Bα − 8
α
+ 6 ω12 − 4ω1 ω2 − 8BE Q1
+6
B
B
Q1
B
AC
AC
2
+ 48
γQ1 + 4 12
− B − 4C ω1 Q31 = 0.
B
B
The equivalence with the Hamilton equations results from the dependence on E
but not on β. However, this type of fourth order first degree ODEs has not yet been
classified, and this would be quite useful to do so, in order to check that no Painlevéintegrable case has been omitted when performing the Painlevé test on the coupled
system made of the two Hamilton equations.
6. Birational transformations between the quartic cases and integrated
equations
Between Hamiltonians with one degree of freedom such as H = p2 /2+aq 2 +bq 3 +q 4
and H = p2 /2 + Aq 2 + q 3 , there exist invertible transformations which allow one
to carry out the solution from one case to the other. These are the well known
homographies between the Jacobi and the Weierstrass elliptic functions. In the present
case of two degrees of freedom, the simplest example of such a transformation is ([15,
Eq. (7.14)])

ω1 + 4ω2
Ω 1 + Ω2
2
2


= αq1 +
,
 Q1 + Q2 +

5
20

α2 2 4ω1 + 26ω2
(ω1 + 4ω2 )2
(26)
(Ω1 − Ω2 )(Q21 − Q22 ) =
q2 −
αq1 −
+ 2E,


2
5
100



Ω1 = ω 1 ,
Ω2 = 4ω2 ,
between the quartic 1:2:1 case H(Qj , Pj , Ω1 , Ω2 , A, B) and the cubic KdV5 case
H(qj , pj , ω1 , ω2 , α, γ). Its action on the hyperelliptic curves is just a translation.
An attempt to find transformations between the other quartic cases and any cubic
case which would be as simple as (26) has been unsuccessful for the moment.
However, it is possible to obtain a birational transformation ([15]) between every
remaining quartic case (1:6:1, 1:6:8, 1:12:16) and some classified fourth order ODE
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
74
R. CONTE, M. MUSETTE & C. VERHOEVEN
of the type (19). Indeed, for each of the seven cases, the two Hamilton equations
are equivalent ([6, 21, 22]) to the traveling wave reduction of a soliton system made
either of a single PDE (HH3) or of two coupled PDEs (HH4), most of them appearing
in lists established from group theory ([19]). Among the various soliton equations
which are equivalent to them via a Bäcklund transformation, some of them admit a
traveling wave reduction to a classified ODE. This property defines a path ([31, 44])
which starts from one of the three remaining HH4 cases, goes up to a soliton system
of two coupled 1+1-dimensional PDEs admitting a reduction to the considered case,
then goes via a Bäcklund transformation to another equivalent 1+1-dim PDE system,
finally goes down by reduction to an already integrated ODE or system of ODEs.
6.1. Integration of the 1:6:1 and 1:6:8 cases with the a-F-VI equation
In this section, the integration is performed via a birational transformation to the
autonomous F-VI equation (a-F-VI) in the classification of Cosgrove ([17]):
α2VI
y + κVI t + βVI , κVI = 0.
9
The two considered Hamiltonians, with their second constant of the motion, are
the following,

Ω
1


H = (P12 + P22 ) + (Q21 + Q22 )


2
2



1
1 κ21
κ22 
4
2 2
4

−
(Q
+
6Q
Q
+
Q
)
−
+
= E,

1
1
2
2
2

32
2 Q1
Q22
1:6:1
(28)
Q2 + Q2
2

1
2

 K = P1 P2 + Q1 Q2 −
+Ω


8


2
2

κ2 κ2
1 2 2
κ
κ



κ1 Q2 + κ22 Q21 + 21 22 ,
− P22 12 − P12 22 +
Q1
Q2
4
Q1 Q2
(27)
2
a-F-VI : y 0000 = 18yy 00 + 9y 0 − 24y 3 + αVI y 2 +
and
(29)

ω
1


H = (p21 + p22 ) + (4q12 + q22 )


2
2


1
β


−
(8q14 + 6q12 q22 + q24 ) − γq1 + 2 = E,



16
2q2


2

2
q
β
1:6:8
K = p22 − 2 (2q22 + 4q12 + ω) + 2

16


q2

1 2

2

− q2 (q2 p1 − 2q1 p2 ) + γ − 2γq22 − 4q2 p1 p2



4


β
1



+ q1 q24 + q13 q22 + 4q1 p22 − 4ωq1 q22 + 4q1 2 .
2
q2
There exists a canonical transformation ([6]) between these two cases, mapping the
constants as follows:
κ1 + κ2
(30) E1:6:8 = E1:6:1 , K1:6:8 = K1:6:1 , ω = Ω, γ =
, β = −(κ1 − κ2 )2 .
2
Therefore, one only needs to integrate either case.
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
75
The path to an integrated ODE comprises the following three segments.
The coordinate q1 (t) of the 1:6:8 case can be identified ([7, 6]) to the component
F of the traveling wave reduction f (x, τ ) = F (x − cτ ), g(x, τ ) = G(x − cτ ) of a soliton
system of two coupled KdV-like equations (c-KdV system) denoted c-KdV1 ([7, 6])

1
3

fτ + fxx + f fx − f 3 + 3f g = 0,



2
2
x
(31)
−2gτ + gxxx + 6ggx + 3f gxx + 6gfxx + 9fx gx − 3f 2 gx



3
3

+ fxxxx + f fxxx + 9fx fxx − 3f 2 fxx − 3f fx2 = 0,
2
2
with the identification
(
q1 = F, q22 = −2 F 0 + F 2 + 2G − 2ω ,
(32)
c = −ω, K1 = γ, K2 = E,
in which K1 and K2 are two constants of integration.
There exists a Bäcklund transformation between this soliton system and another
one of c-KdV type, denoted bi-SH system ([19]):
(
−2uτ + uxx + u2 + 6v x = 0,
(33)
vτ + vxxx + uvx = 0.
This Bäcklund transformation is defined by the Miura transformation ([31])

3

 u=
2g − fx − f 2 ,
2
(34)

 v = 3 2fxxx + 4f fxx + 8gfx + 4f gx + 3fx2 − 2f 2 fx − f 4 + 4gf 2 .
4
Finally, the traveling wave reduction
u(x, τ ) = U (x − cτ ),
v(x, τ ) = V (x − cτ )
can be identified ([44]) to the autonomous F-VI equation (a-F-VI) (27), whose general solution is meromorphic, expressed with genus two hyperelliptic functions ([17,
Eq. (7.26)]). The identification is

c

U = −6 y +
,


18

16
KA
4
(35)
,
V = y 00 − 6y 2 + cy + c2 −

3
27
2


512
 α = −4c, β = K − 2cK +
c3 ,
VI
VI
B
A
243
in which KA , KB are two constants of integration.
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
76
R. CONTE, M. MUSETTE & C. VERHOEVEN
In order to perform the integration of both the 1:6:1 and the 1:6:8 cases, it is
sufficient to express (F, G) rationally in terms of (U, V, U 0 , V 0 ). The result is

W0
K1 2

F =
+
− 3U 0 − 2(U − 3c) 12V + (U + 3c)2 + 36KB − 54K12 ,



2W
24W




1
U


 G =
(2V + 3K2 ) 2V 00 + K1 U 0 − 3K12
+
3
8W
(36)
− 2(U − 3c) 2K1 V 0 + K12 (U + 3c) ,




3 2 3 2


K2 + K1 (U − 3c),
W
=
V
+


2
2



KA = K2 .
Making the product of the successive transformations (32), (36), (35), one obtains
a meromorphic general solution for q1 , q22 :

4
γ
9 2
W0


9j − 3 y + ω (h + E) − γ ,
+
q1 =



2W
W
9
4






5
1 h 0 γ 2

2

q
=
−16
y
−
− 48y 3 − 16ωy 2
12 y +
ω
+

2


9
W
2




128 2
1280 3 40
3


+ 24E +
ω y+
ω − ωE + β



9
243
3
4

2 i



5
5


− 24γ y − ω h0 − 144γ 2 y − ω
,
9
9
(37)




5

2
2

y
−
ω
,
W
=
(h
+
E)
−
9γ


9




3
512 3
3


ω ,
αVI = 4ω,
βVI = γ 2 + 2ωE − β −



4
16
243



3
1
3
1
9


 K1,VI =
K − E2,
K2,VI =
EK − E 3 + βγ 2 ,


32
2
32
3
64




3
3


K1 = γ,
K2 = E,
KA = E,
KB = − β + γ 2 ,
16
4
in which h and



y









(38)

h









 j
j are the convenient auxiliary variables ([17, Eqs. (7.4)–(7.5)])
p
Q(s1 , s2 ) + Q(s1 )Q(s2 )
5
= p
2 + αVI ,
p
36
2
s21 − CVI + s22 − CVI
p
3
s1 s2 + CVI + (s21 − CVI )(s22 − CVI ) FVI
= − EVI
−
,
4
s1 + s2
2
αVI
EVI
1
−
.
= (2h + FVI ) y +
6
9
4(s1 + s2 )
SÉMINAIRES & CONGRÈS 14
HÉNON-HEILES HAMILTONIANS
77
In the above, the variables s1 , s2 are defined by the hyperelliptic system ([17])

p
p

(s1 − s2 )s01 = P (s1 ),
(s2 − s1 )s02 = P (s2 ),





 P (s) = (s2 − CVI )Q(s),
(39)
EVI
αVI 2


(s + t2 − 2CVI ) +
(s + t) + FVI ,
Q(s, t) = (s2 − CVI )(t2 − CVI ) −


2
2



Q(s) = Q(s, s).
Despite their square roots, the symmetric expressions in (38) are nevertheless meromorphic ([20, 30]).
The completeness of both the 1:6:1 and 1:6:8 Hamiltonians results from the completeness of the a-F-VI ODE and the following counting. The 1:6:8 depends on the
parameters (ω, β, γ, E, K), the a-F-VI ODE and its hyperelliptic system depend on
the same number of parameters (α, β, C, E, F )VI , and these two sets of five parameters
are linked by exactly five algebraic relations ([17, Eqs. (7.9)-(7.12)]):

αVI = 4ω,






3
3
512 3


βVI = γ 2 + 2ωE − β −
ω ,



4
16
243



16
2
(40)
EVI
= − ω(FVI − 2E) − β + 4γ 2 ,

3




4 2

2

CVI EVI
= (FVI
− 4E 2 ) + K,



3




27
2
(FVI − 2E)2 (FVI + 4E) + 9K
4 (FVI − 2E) − 4 βγ = 0.
The algebraic nature (instead of rational like in the 1:2:1 case and the three cubic
cases) of these dependence relations could explain the difficulty to separate the variables in the Hamilton-Jacobi equation. In the nongeneric case βγ = 0, i.e, κ21 = κ22 ,
for which the separating variables are known ([34]), the coefficients (α, β, C, E 2 , F )VI
become rational functions of (ω, β, γ, E, K), see [43]. Since these separating variables
have been obtained by the same method as in the cubic SK-KK case, it would be
quite useful to remove the difficulty which remains in the method based on Laurent
series, see Section 3.
6.2. Integration of the 1:12:16 case by a birational transformation. — This
is the only case for which the integration, which can indeed be performed with the
same results (meromorphy of the general solution, completeness of the Hamiltonian)
is not satisfying. Indeed, the hyperelliptic system to which the 1:12:16 has been
mapped by a birational transformation ([15]) is essentially different from the hyperelliptic system resulting from the separating variables ([41]) in the nongeneric case
αβ = 0 for which they are known. Since the nongeneric subcase α = 0 belongs to the
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
R. CONTE, M. MUSETTE & C. VERHOEVEN
78
Stäckel class (two invariants quadratic in p1 , p2 ), for which the separating variables
are unambiguous, this indicates that some progress has still to be made.
The main remarkable feature of the 1:12:16 is the existence of a twin system to
which it is mapped by a canonical transformation ([6, 7]) which only differs by numerical coefficients from the canonical transformation between the cubic SK and KK
cases. The two systems are the following ones:
(41) 1 : 12 : 16



H














K








=
=
1 2
Ω
(P + P22 ) + (4Q21 + Q22 )
2 1
8
1 κ21
4κ22 1
4
= E,
+
− (16Q1 + 12Q21 Q22 + Q42 ) −
32
2 Q21
Q22
1
8(Q2 P1 − Q1 P2 )P2 − Q1 Q42 − 2Q31 Q22
16
Q2 P 2
κ2 2
+2ΩQ1 Q22 + 32Q1 22 + κ21 Q42 − 4 2 2 2 .
Q2
Q1
and [this system is not the sum of a kinetic energy and a potential energy]
(42)

2 1
1 2 3


H
=
q
q
p
+
p
−
− (4q14 + 9q12 q22 + 5q24 )
1 2
2

1

2
2
8



ζ
ω 2


(q1 + q22 ) − κq1 + 2 = E,
+


2
2q

2



1

2
2 2
4
2
2

K = 2 (2q2 p1 + 2q1 q2 − 2q1 q2 p2 − q2 − 4κq1 ) 2q2 p1 + 2q12 q22


q2




p2
κ2


+p22 − 4q1 q2 p2 − 2q24 + Ωq22 + 4 2 + 8κq1 − 4κ
q
q2
5:9:4
2


p2
κ


+4(ζ + 4κ2 )
− 2q1 + 4q12 + q22 + 4q1 2 p1



q2
q2



2

2
3
q
1



− 4 (q12 q22 + q24 + 2κq1 )2 + 2 12 p2 − q1 q2


q2
q2
2






(q 2 + q22 )2
ζ


+ 1
+ q12 4 .

2
q2
The canonical transformation maps the constants as follows:
(43) E5:9:4 = E1:12:16 , K5:9:4 = K1:12:16 , ω = Ω, κ =
κ1 + κ2
, ζ = −(κ1 − κ2 )2 .
2
The path to an integrated ODE is quite similar to that described in detail in
section 6.1, in particular it is also made of three segments ([6, 31, 41]). The result
is the following ([15]):
(44)
SÉMINAIRES & CONGRÈS 14
Q1 , Q22 = rational(y, y 0 , y 00 , y 000 ),
HÉNON-HEILES HAMILTONIANS
79
in which y obeys the F-IV equation in the classification of Cosgrove ([17]),
 0000
00
3

 y = 30yy − 60y + αIV y + βIV ,





1 0


s1 + s02 + s21 + s1 s2 + s22 + A ,
y =


2



p
p


(s1 − s2 )s01 = P (s1 ),
(s2 − s1 )s02 = P (s2 ),
(45)
F-IV



βIV
αIV 2


(s + A) + Bs +
,
P (s) = (s2 + A)3 −


3
3




2


9AB 2
3B


 K1,IV =
,
K2,IV = −
4
64
with (K1,IV , K2,IV ) two polynomial first integrals of F-IV. The general solution of this
ODE is meromorphic, expressed with genus two hyperelliptic functions ([17]). This
proves the PP for the 1:12:16.
In the two nongeneric cases κ1 κ2 = 0 where the separating variables are known,
the hyperelliptic curve is
K
s + κ21 + κ22 = 0,
20
and it does not coincide in this case with the hyperelliptic curve of F-IV. Therefore, F-IV (as well as its birationally equivalent ODE F-III) is not the good ODE
to consider, and it should be quite instructive to directly integrate the fourth order
equivalent ODE (26) in that case.
(46)
κ1 κ2 = 0 : P (s) = s6 − ωs3 + 2Es2 +
7. Conclusion and open problems
All the time independent two-degree-of-freedom Hamiltonians which possess the
Painlevé property have a meromorphic general solution, expressed with hyperelliptic
functions of genus two. Moreover, all such Hamiltonians are complete in the Painlevé
sense, i.e, it is impossible to add any term to the Hamiltonian without ruining the
Painlevé property.
As to the remaining open problems, depending on the center of interest, they are
1. from the point of view of Hamiltonian theory, one has to find the separating
variables in the three missing quartic cases. This should be possible by the
methods of Sklyanin and van Moerbeke and Vanhaecke;
2. from the point of view of the integration of differential equations, the problem
remains to enumerate all the fourth order first degree differential equations in a
given precise class which possess the Painlevé property.
SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2006
80
R. CONTE, M. MUSETTE & C. VERHOEVEN
Let us finally mention that the time dependent extension of these seven cases has
been studied in Refs. [27, 28].
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R. Conte, Service de physique de l’état condensé (U.R.A. 2464), C.E.A. Saclay, F-91191 Gif-surYvette Cedex, France • E-mail : Robert.Conte@cea.fr
M. Musette, Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel • International
Solvay Institutes for Physics and Chemistry, Pleinlaan 2, B–1050 Brussels, Belgium
E-mail : MMusette@vub.ac.be
C. Verhoeven, Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel • International
Solvay Institutes for Physics and Chemistry, Pleinlaan 2, B–1050 Brussels, Belgium
E-mail : CVerhoev@vub.ac.be
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